# Trace formula for the magnetic Laplacian

## TL;DR

This paper surveys the Guillemin-Uribe trace formula for the magnetic Laplacian, illustrating how it connects magnetic geodesic flow dynamics with eigenvalues, and provides explicit examples on specific surfaces.

## Contribution

It offers a comprehensive overview of the Guillemin-Uribe trace formula and computes explicit examples for certain magnetic surfaces, enhancing understanding of magnetic Laplacian spectra.

## Key findings

- Explicit trace formula computations for constant curvature surfaces.
- Illustrations of magnetic geodesic flow dynamics.
- Connections between eigenvalues and magnetic fields.

## Abstract

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. In this paper, we give a survey of basic notions and results related with the Guillemin-Uribe trace formula and provide concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example.